XYZ Corporation produces a commercial product that is in great demand by consumers on a national basis. Unfortunately, near the plant where it is produced there is a large population of dove-tail turtles who are adversely affected by contaminants from the plant. XYZ has a filtering process that is expensive and any increase in filtering effectiveness reduces their profit. Dove tailed turtles are not a protected species hence there are no environmental rules regulating XYZs level of contaminants.

Clearly, no filtering at all would maximize XYZs profitability. However, a local environmental group monitors XYZs contaminant level and maintains a website showing the percentage mortality rate of the dove tailed turtles due to XYZs contaminants. XYZ has noticed that the higher the mortality percentage, the less items are bought and the lower their profitability.

Their Marketing Department and Research Group has established the following Revenue Function, R(x), as a function of Dove Tail Turtle Mortality expressed as decimal between 0 to 1, representing mortality rate:

R(x) = 1 + x x²; 0 < x 1

R(x) is expressed in billions of dollars and represents the revenue generated. Since XYZ has fixed operating costs of one billion dollars, the profit function, P(x) is given by P(x) = R(x) 1.

You have been hired as the mathematical consultant for XYZ Corporation. They have asked you to help them optimize their profit and request that you:

1. Find the optimal dove-tail turtle mortality rate percentage that will maximize revenue.

2. State what the maximum profit will be.